On perfect powers that are sums of cubes of a five term arithmetic progression
نویسندگان
چکیده
منابع مشابه
Perfect Powers That Are Sums of Consecutive Cubes
Euler noted the relation 63= 33+43+53 and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular, Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker’s work by determi...
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We show that the abc conjecture implies that the number of terms of any arithmetic progression consisting of almost perfect ”inhomogeneous” powers is bounded, moreover, if the exponents of the powers are all ≥ 4, then the number of such progressions is finite. We derive a similar statement unconditionally, provided that the exponents of the terms in the progression are bounded from above.
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Let n ≥ 3. This paper is concerned with the equation a3 + b3 = cn, which we attack using a combination of the modular approach (via Frey curves and Galois representations) with obstructions to the solutions that are of Brauer–Manin type. We shall show that there are no solutions in coprime, non-zero integers a, b, c, for a set of prime exponents n having Dirichlet density 28219 44928 ≈ 0.628, a...
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1 Background The problem of cubes that are sums of consecutive cubes goes back to Euler ([10] art. 249) who noted the remarkable relation 33 + 43 + 53 = 63. Similar problems were considered by several mathematicians during the nineteenth and early twentieth century as surveyed in Dickson’sHistory of the Theory of Numbers ([7] p. 582–588). These questions are still of interest today. For example...
متن کاملOn Powers as Sums of Two Cubes
In a paper of Kraus, it is proved that x 3 + y 3 = z p for p 17 has only trivial primitive solutions, provided that p satisses a relatively mild and easily tested condition. In this article we prove that the primitive solutions of x 3 + y 3 = z p with p = 4; 5; 7; 11; 13, correspond to rational points on hyperelliptic curves with Jaco-bians of relatively small rank. Consequently, Chabauty metho...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2019
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2019.02.014